Well, European Christian states did tried to help Byzantines by fighting the Ottomans before but they were severely defeated in Battle of Varna in 1444, where join armies of Poland, Lithuania, Hungary, Papal States, Wallachia, Moldavia with various auxiliaries were defeated by Murad II (father of Mehmed II, who later conquered Constantinople). The defeat was very heavy, King Ladislaus who was monarch of Poland, Lithuania and Hungary was killed along with many noble knights and the papal legate. It is safe to assume that 8 years later that most Christian nations still have difficulty regaining will and strength to fight the Ottomans.

Having said that, the Italian city-states of Genoa and Venice did tried to help the Byzantines during the siege but their navy faced the Ottoman navy and the city also choked by twin fortress of Rumelihisari-Anadoluhisari, making reinforcement and supply very difficult. A few Genoese soldiers that did arrived would later be killed by the Ottomans which heavily outnumbered them.